Existence and uniqueness of optimal transport maps in locally compact CAT(0) spaces

2023/03/03 by Bërdëllima, Arian
#28A50 #46E27 #49Q15 #49Q22 #51F30 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.2303.02082

Abstract

We show that in a locally compact complete CAT(0) space satisfying positive angles property and a disintegration regularity for its canonical Hausdorff measure, there exists a unique optimal transport map that push-forwards a given absolutely continuous probability measure to another probability measure. In particular this holds for the Riemannian manifolds of non-positive sectional curvature and CAT(0) Euclidean polyhedral complexes. Moveover we give a polar factorization result for Borel maps in CAT(0) spaces in terms of optimal transport maps and measure preserving maps.

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